Whether you use online or the download, you will receive an email after you e-file your return.
When you e-file your federal return you will receive two emails from TurboTax. The first one will say that your return was submitted. The second email will tell you if your federal return was accepted or rejected. If you e-filed a state return, there will be a third email to tell you if the state accepted or rejected your state return.
You can launch your downloaded program and check your filing status --- Click on File and then click on Check E-File Status.